In a certain test, there are n questions. In this, `2^( n-r )` students gave wrong answers to at least `(r)` questions, where `r=0,1,2,….n`. If the total number of wrong answer is 4095, then the value of n is

Options

A. 11
B. 12
C. 13
D. 15

Answer

To solve the problem step by step, we will analyze the information given and derive the value of \( n \).

### Step 1: Understanding the Problem
We know that for \( n \) questions, the number of students who gave wrong answers to at least \( r \) questions is given by \( 2^{n-r} \). We need to find \( n \) such that the total number of wrong answers is 4095.

### Step 2: Calculate Total Wrong Answers
The total number of wrong answers can be calculated by summing the number of students who gave wrong answers for each value of \( r \) from 0 to \( n \):
- For \( r = 0 \): \( 2^n \) students gave wrong answers to at least 0 questions.
- For \( r = 1 \): \( 2^{n-1} \) students gave wrong answers to at least 1 question.
- For \( r = 2 \): \( 2^{n-2} \) students gave wrong answers to at least 2 questions.
- ...
- For \( r = n \): \( 2^{n-n} = 2^0 = 1 \) student gave wrong answers to at least \( n \) questions.

The total number of wrong answers can be expressed as:
\[ \text{Total wrong answers} = 2^n + 2^{n-1} + 2^{n-2} + \ldots + 2^0 \]

### Step 3: Sum of the Geometric Series
The expression \( 2^n + 2^{n-1} + 2^{n-2} + \ldots + 2^0 \) is a geometric series. The sum of a geometric series can be calculated using the formula:
\[ S = a \frac{(r^n - 1)}{(r - 1)} \]
where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

In our case:
- \( a = 2^0 = 1 \)
- \( r = 2 \)
- The number of terms is \( n + 1 \) (from \( r = 0 \) to \( r = n \))

Thus, the sum becomes:
\[ S = 1 \cdot \frac{(2^{n+1} - 1)}{(2 - 1)} = 2^{n+1} - 1 \]

### Step 4: Set Up the Equation
According to the problem, the total number of wrong answers is 4095. Therefore, we can set up the equation:
\[ 2^{n+1} - 1 = 4095 \]

### Step 5: Solve for \( n \)
To solve for \( n \), we first add 1 to both sides:
\[ 2^{n+1} = 4096 \]

Next, we recognize that \( 4096 \) is a power of 2:
\[ 4096 = 2^{12} \]

Thus, we have:
\[ 2^{n+1} = 2^{12} \]

Since the bases are the same, we can equate the exponents:
\[ n + 1 = 12 \]

Finally, solving for \( n \):
\[ n = 12 - 1 = 11 \]

### Conclusion
The value of \( n \) is \( 11 \).